Classification of Finite Rings of Order p 6 by Generators and Relations

Abstract

For any finite abelian group ( R , + ) , we define a binary operation or "multiplication" on R and give necessary and sufficient conditions on this multiplication for R to extend to a ring. Then we show when two rings made on the same group are isomorphic. In particular, it is shown that there are n + 1 rings of order [superscript] p n [/superscript] with characteristic [superscript] p n [/superscript] , where p is a prime number. Also, all finite rings of order [superscript] p 6 [/superscript] are described by generators and relations. Finally, we give an algorithm for the computation of all finite rings based on their additive group.

P. Karimi Beiranvand 1 and R. Beyranvand 2 and M. Gholami 3

Academic Editor:Liwei Zhang

1, Department of Mathematics, Islamic Azad University, Khorramabad Branch, Khorramabad 68178-16645, Iran
2, Department of Mathematics, Lorestan University, P.O. Box 465, Khorramabad 68137-17133, Iran
3, Department of Mathematical Sciences, Shahrekord University, P.O. Box 115, Shahrekord 88186-34141, Iran

Received 26 September 2012; Accepted 10 December 2012

This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction

The problem of determining and classifying up to isomorphism finite rings has received considerable attention, both old and new, see [1-4]. It is well known that if [figure omitted; refer to PDF] is a finite ring (with identity), then the additive group of [figure omitted; refer to PDF] splits as the direct of its [figure omitted; refer to PDF] -primary components [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] is a prime number, and these are pairwise orthogonal ideals. Thus [figure omitted; refer to PDF] is the direct sum of the rings [figure omitted; refer to PDF] . We provide a new proof of this fact in the paper.

Now let [figure omitted; refer to PDF] be a prime number. It is easy to see that there are two rings of order [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and the null ring of order [figure omitted; refer to PDF] . In [5], Raghavendran (1969) proved that there exist eleven rings of order [figure omitted; refer to PDF] , only four of these rings have identity. Also in [6], Gilmer and Mott (1973) showed that there exist [figure omitted; refer to PDF] rings of order [figure omitted; refer to PDF] , only twelve of these rings have identity, and 59 rings of order 8, only eleven of these rings have identity. For [figure omitted; refer to PDF] , a comprehensive list of noncommutative rings was first only drawn up in [7]. Commutative rings of order [figure omitted; refer to PDF] have been characterized by Wilson [4]. Finally, Corbas and Williams (2000) in [1, 8], determined all rings of order [figure omitted; refer to PDF] . The rings of upper order are not still characterized.

Throughout the paper, all rings are associative (not necessarily commutative or with identity). For a set [figure omitted; refer to PDF] , [figure omitted; refer to PDF] denotes the cardinal of [figure omitted; refer to PDF] . For two integers [figure omitted; refer to PDF] and positive integer [figure omitted; refer to PDF] , we denote [figure omitted; refer to PDF] in case [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are congruent modulo [figure omitted; refer to PDF] , and [figure omitted; refer to PDF] in case [figure omitted; refer to PDF] divides [figure omitted; refer to PDF] . Also, for two integers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , gcd [figure omitted; refer to PDF] denotes the greatest common divisor of [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . In [9], the authors introduced a multiplication (similar to Theorem 3) on a finitely generated [figure omitted; refer to PDF] -module to extend to an [figure omitted; refer to PDF] -algebra, where [figure omitted; refer to PDF] is a commutative ring. Also a number of necessary and sufficient conditions for any [figure omitted; refer to PDF] -module to extend to an [figure omitted; refer to PDF] -algebra, where [figure omitted; refer to PDF] is a commutative ring, is given by Behboodi et al. [9]. Here instead of modules over a commutative ring, we focus our attention to the [figure omitted; refer to PDF] -modules. In Section 2, by using the method in [9], we define a binary operation or "multiplication" on a finite abelian group and extend to a ring. In particular, we prove that there are [figure omitted; refer to PDF] rings (up to isomorphism) of order [figure omitted; refer to PDF] whose abelian group is cyclic. Also, all rings of order [figure omitted; refer to PDF] whose abelian group is [figure omitted; refer to PDF] are determined. In Section 3, an algorithm for the computation of all finite rings based on their additive group is given.

2. A Representation of Finite Rings and Fundamental Theorems

We begin the paper with the following well known fact and give a new proof of this fact.

Theorem 1.

Let [figure omitted; refer to PDF] be a ring of order [figure omitted; refer to PDF] , where the [figure omitted; refer to PDF] are distinct primes and the [figure omitted; refer to PDF] are positive integers. Then [figure omitted; refer to PDF] is expressible, in a unique manner, as the direct sum of rings [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Proof.

We know that the additive group [figure omitted; refer to PDF] is expressible, in a unique manner, as the direct sum of groups [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . We claim that [figure omitted; refer to PDF] is an ideal of [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . Suppose that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] + [figure omitted; refer to PDF] + [figure omitted; refer to PDF] + [figure omitted; refer to PDF] + [figure omitted; refer to PDF] + [figure omitted; refer to PDF] . Since [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] , [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] . If [figure omitted; refer to PDF] , then we must have [figure omitted; refer to PDF] , a contradiction. Thus [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] . Now suppose that [figure omitted; refer to PDF] and [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . We show that [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] for [figure omitted; refer to PDF] . Now for any [figure omitted; refer to PDF] , if [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] , a contradiction. Thus for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . On the other hand, since [figure omitted; refer to PDF] , by a similar argument, we conclude that for all [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] and so for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is an ideal of [figure omitted; refer to PDF] , and the proof is complete.

Remark 2.

If [figure omitted; refer to PDF] is a finite ring, then its additive group is a finite abelian group and is thus a direct product of cyclic groups. Suppose these have generators [figure omitted; refer to PDF] of orders [figure omitted; refer to PDF] . Then the ring structure is determined by the [figure omitted; refer to PDF] products [figure omitted; refer to PDF] and thus by the [figure omitted; refer to PDF] structure constants [figure omitted; refer to PDF] . As [6] we introduce a convenient notation, motivated by group theory, for giving the structure of a finite ring. A presentation for a finite ring [figure omitted; refer to PDF] consists of a set of generators [figure omitted; refer to PDF] of the additive group of [figure omitted; refer to PDF] together with relations. The relations are of two types:

(i) [figure omitted; refer to PDF] for [figure omitted; refer to PDF] ;

(ii) [figure omitted; refer to PDF] with [figure omitted; refer to PDF] for [figure omitted; refer to PDF] .

If the ring [figure omitted; refer to PDF] has the presentation above we write [figure omitted; refer to PDF]

Theorem 3.

Let [figure omitted; refer to PDF] be a finite ring with additive group [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] [figure omitted; refer to PDF] are cyclic subgroups of orders [figure omitted; refer to PDF] , respectively. Assume [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] , and " [figure omitted; refer to PDF] " is the following operation [figure omitted; refer to PDF] Then

(1) " [figure omitted; refer to PDF] " is well-defined if and only if [figure omitted; refer to PDF]

(2) " [figure omitted; refer to PDF] " is always distributive, in the case when " [figure omitted; refer to PDF] " is well-defined;

(3) " [figure omitted; refer to PDF] " is associative if and only if [figure omitted; refer to PDF]

Consequently, [figure omitted; refer to PDF] is a ring if and only if [figure omitted; refer to PDF] and [figure omitted; refer to PDF] above hold.

Proof.

(1) ( [figure omitted; refer to PDF] ). Assume " [figure omitted; refer to PDF] " is well-defined and fix [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] . Then by definition of " [figure omitted; refer to PDF] ", we have [figure omitted; refer to PDF] Thus for each [figure omitted; refer to PDF] , we deduce [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . Similarly, [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . Therefore, [figure omitted; refer to PDF] Now one can easily check that the last relation holds if and only if [figure omitted; refer to PDF] holds.

(1) ( [figure omitted; refer to PDF] ). Assume [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] , [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Hence [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] Now by condition [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] that is, [figure omitted; refer to PDF] Thus the operation " [figure omitted; refer to PDF] " is well-defined.

(2) Suppose the operation " [figure omitted; refer to PDF] " is well-defined. Then [figure omitted; refer to PDF]

On the other hand [figure omitted; refer to PDF] Thus " [figure omitted; refer to PDF] " is a distributive operation.

(3) By definition of " [figure omitted; refer to PDF] ", we have [figure omitted; refer to PDF] and so [figure omitted; refer to PDF] and also [figure omitted; refer to PDF] Now, it is clear that " [figure omitted; refer to PDF] " is associative if and only if [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , if and only if [figure omitted; refer to PDF]

Theorem 4.

For [figure omitted; refer to PDF] , let [figure omitted; refer to PDF] be two presentations with suitable [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Assume [figure omitted; refer to PDF] where [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the following map: [figure omitted; refer to PDF] Then

(1) [figure omitted; refer to PDF] is well-defined if and only if for [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

(2) If [figure omitted; refer to PDF] is well-defined, then

(a) [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -module homomorphism,

(b) [figure omitted; refer to PDF] is one to one if and only if for each [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , and

(c) [figure omitted; refer to PDF] is a ring homomorphism if and only if for [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Proof.

(1) ( [figure omitted; refer to PDF] ). Assume [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] . Then by definition [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] It follows that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . Therefore, [figure omitted; refer to PDF] , for all [figure omitted; refer to PDF] and so for [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

(1) ( [figure omitted; refer to PDF] ). Suppose [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] [figure omitted; refer to PDF] ) and so [figure omitted; refer to PDF] . Now by our hypothesis [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] . Therefore, [figure omitted; refer to PDF] This means that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] is well-defined.

(2) (a) Assume [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] . Clearly, [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] is an [figure omitted; refer to PDF] -module homomorphism.

(2) (b) By definition [figure omitted; refer to PDF] , (b) is clear.

(2) (c) By (2) (a), [figure omitted; refer to PDF] is always a group homomorphism. Thus it is sufficient to show that [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] for each [figure omitted; refer to PDF] . However by multiplications of [figure omitted; refer to PDF] , [figure omitted; refer to PDF] and definition of [figure omitted; refer to PDF] we have [figure omitted; refer to PDF] and also [figure omitted; refer to PDF] Thus [figure omitted; refer to PDF] is a ring homomorphism if and only if for all [figure omitted; refer to PDF] , [figure omitted; refer to PDF]

Corollary 5.

Let [figure omitted; refer to PDF] be a positive integer and let [figure omitted; refer to PDF] [figure omitted; refer to PDF] be cyclic additive groups of orders [figure omitted; refer to PDF] , respectively. If [figure omitted; refer to PDF] are two presentations with suitable [figure omitted; refer to PDF] [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] if and only if there exist [figure omitted; refer to PDF] ( [figure omitted; refer to PDF] ) such that

(1) [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] ;

(2) for all [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] for all [figure omitted; refer to PDF] , and

(3) [figure omitted; refer to PDF] , for [figure omitted; refer to PDF] .

Proof.

It is clear by Theorem 4.

3. Finite Rings of Order [figure omitted; refer to PDF] with Characteristic [figure omitted; refer to PDF] and [figure omitted; refer to PDF]

In this section, we first classify all finite rings of order [figure omitted; refer to PDF] with characteristic [figure omitted; refer to PDF] . Then this classification is extended to the rings order [figure omitted; refer to PDF] with characteristic [figure omitted; refer to PDF] .

Proposition 6.

(i) Let [figure omitted; refer to PDF] be an additive cyclic group of order [figure omitted; refer to PDF] and [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] [figure omitted; refer to PDF] [figure omitted; refer to PDF] is always a ring.

(ii) Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two presentations with [figure omitted; refer to PDF] . Assume [figure omitted; refer to PDF] and [figure omitted; refer to PDF] is the following map: [figure omitted; refer to PDF] Then

(1) [figure omitted; refer to PDF] is well-defined;

(2) [figure omitted; refer to PDF] is a ring homomorphism if and only if for [figure omitted; refer to PDF] ;

(3) [figure omitted; refer to PDF] is one to one if and only if for each [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] .

Proof.

(i) It is straightforward by relations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Theorem 3.

(ii) The proof is obtained by simplification of the relations in Theorem 4.

Since the function introduced in preceding proposition is multiplication by some [figure omitted; refer to PDF] , and is always well-defined, we have the following corollary.

Corollary 7.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two presentations with [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] if and only if there exists [figure omitted; refer to PDF] such that

(1) [figure omitted; refer to PDF] ;

(2) for each [figure omitted; refer to PDF] implies that [figure omitted; refer to PDF] .

Proof.

It is clear by Proposition 6.

Theorem 8.

For any prime number [figure omitted; refer to PDF] , there are exactly seven rings of order [figure omitted; refer to PDF] whose additive group is cyclic, only one of these rings have identity. However all of these are commutative.

Proof.

Let [figure omitted; refer to PDF] and [figure omitted; refer to PDF] be two presentations with [figure omitted; refer to PDF] . The relation [figure omitted; refer to PDF] in Corollary 7 implies that [figure omitted; refer to PDF] is relatively prime to [figure omitted; refer to PDF] and so by the relation [figure omitted; refer to PDF] in Corollary 7, we deduce that [figure omitted; refer to PDF] . Since the relation [figure omitted; refer to PDF] has a solution, gcd [figure omitted; refer to PDF] . Now we proceed by cases.

Case 1. gcd [figure omitted; refer to PDF] , that is, [figure omitted; refer to PDF] is relatively prime to [figure omitted; refer to PDF] . Since gcd [figure omitted; refer to PDF] , [figure omitted; refer to PDF] is also relatively prime to [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] where both [figure omitted; refer to PDF] and [figure omitted; refer to PDF] are relatively prime to [figure omitted; refer to PDF] .

Case 2. gcd [figure omitted; refer to PDF] . Then the relation [figure omitted; refer to PDF] shows that either gcd [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . But the relation [figure omitted; refer to PDF] follows that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] because gcd [figure omitted; refer to PDF] . Since gcd [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Therefore we obtain that [figure omitted; refer to PDF] and so gcd [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] where both gcd [figure omitted; refer to PDF] and gcd [figure omitted; refer to PDF] .

Case 3. gcd [figure omitted; refer to PDF] . Then the relation gcd [figure omitted; refer to PDF] shows that either gcd [figure omitted; refer to PDF] or [figure omitted; refer to PDF] . But the relation [figure omitted; refer to PDF] follows that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] because gcd [figure omitted; refer to PDF] . Since gcd [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Therefore we obtain that [figure omitted; refer to PDF] and so gcd [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] where both gcd [figure omitted; refer to PDF] and gcd [figure omitted; refer to PDF] .

Case 4. gcd [figure omitted; refer to PDF] . Then the relation gcd [figure omitted; refer to PDF] shows that either gcd [figure omitted; refer to PDF] or [figure omitted; refer to PDF] But the relation [figure omitted; refer to PDF] follows that [figure omitted; refer to PDF] and hence [figure omitted; refer to PDF] because gcd [figure omitted; refer to PDF] . Since gcd [figure omitted; refer to PDF] , [figure omitted; refer to PDF] . Therefore we obtain that [figure omitted; refer to PDF] and so gcd [figure omitted; refer to PDF] . Thus [figure omitted; refer to PDF] where both gcd [figure omitted; refer to PDF] and gcd [figure omitted; refer to PDF] .

Case 5. gcd [figure omitted; refer to PDF] . By a similar argument, we deduce [figure omitted; refer to PDF] where both gcd [figure omitted; refer to PDF] and gcd [figure omitted; refer to PDF] .

Case 6. gcd [figure omitted; refer to PDF] . By a similar argument, we deduce [figure omitted; refer to PDF] where both gcd [figure omitted; refer to PDF] and gcd [figure omitted; refer to PDF] .

Case 7. gcd [figure omitted; refer to PDF] . By a similar argument, we deduce [figure omitted; refer to PDF] where both gcd [figure omitted; refer to PDF] and gcd [figure omitted; refer to PDF] .

An analysis similar to that in the proof of Theorem 8 shows that there are exactly [figure omitted; refer to PDF] rings of order [figure omitted; refer to PDF] with characteristic [figure omitted; refer to PDF] . These rings are [figure omitted; refer to PDF]

[figure omitted; refer to PDF]

We can obtain the following result by applying relations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Theorem 3, for the rings of order [figure omitted; refer to PDF] with characteristic [figure omitted; refer to PDF] . Also, this manner may be used for obtaining all finite rings of order [figure omitted; refer to PDF] with characteristic [figure omitted; refer to PDF] , where [figure omitted; refer to PDF] .

Corollary 9.

Let [figure omitted; refer to PDF] where A₁ = <a₁ > and A₂ = <a₂ > be cyclic groups of order [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively. Assume [figure omitted; refer to PDF] where [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] is a ring if and only if the following hold.

(1) [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ;

(2) [figure omitted; refer to PDF] ;

(3) [figure omitted; refer to PDF] ;

(4) [figure omitted; refer to PDF] ;

(5) [figure omitted; refer to PDF] ;

(6) [figure omitted; refer to PDF] .

Application and simplification of the relations (1), (2), and (3) in Corollary 5, for the rings of order [figure omitted; refer to PDF] with characteristic [figure omitted; refer to PDF] , give us the following result.

Corollary 10.

Let [figure omitted; refer to PDF] be two rings with suitable integers [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfied in Theorem 3, for [figure omitted; refer to PDF] . Then [figure omitted; refer to PDF] if and only if there exist [figure omitted; refer to PDF] such that the following hold.

(1) [figure omitted; refer to PDF] ;

(2) If [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , then [figure omitted; refer to PDF] and [figure omitted; refer to PDF] ;

(3) [figure omitted; refer to PDF] ;

(4) [figure omitted; refer to PDF] ;

(5) [figure omitted; refer to PDF] ;

(6) [figure omitted; refer to PDF] ;

(7) [figure omitted; refer to PDF] ;

(8) [figure omitted; refer to PDF] ;

(9) [figure omitted; refer to PDF] .

In the following, one gives an algorithm for computing the finite rings whose abelian group is isomorphic to [figure omitted; refer to PDF] . We represent the algorithm in the form of two sub-algorithms. In Algorithm 11, one lets [figure omitted; refer to PDF] be the input and [figure omitted; refer to PDF] be the set of all sequences [figure omitted; refer to PDF] that satisfy the condition [figure omitted; refer to PDF] , for any [figure omitted; refer to PDF] . Then one has the sequences [figure omitted; refer to PDF] that satisfy the conditions [figure omitted; refer to PDF] and [figure omitted; refer to PDF] in Theorem 3 as the outputs of the algorithm. In fact, any sequence [figure omitted; refer to PDF] that is an output of this algorithm shows a ring as the following: [figure omitted; refer to PDF] In Algorithm 12, one represents the conditions required for the isomorphism of two rings based upon Theorem 4. In fact, one verifies the two sequences [figure omitted; refer to PDF] and [figure omitted; refer to PDF] obtained from Algorithm 11, under isomorphic conditions in Theorem 4. Therefore under Algorithms 11 and 12, one computes all the rings whose abelian group is isomorphic to [figure omitted; refer to PDF] and are not isomorphic. We have implemented Algorithms 11 and 12 in [figure omitted; refer to PDF] and presents a sample of its responses for the rings whose abelian group is isomorphic to [figure omitted; refer to PDF] in the following example.

Algorithm 11.

(1) Input [figure omitted; refer to PDF]

(2) Let [figure omitted; refer to PDF]

(3) For all [figure omitted; refer to PDF] , let [figure omitted; refer to PDF]

(4) Let [figure omitted; refer to PDF] ; for all [figure omitted; refer to PDF]

(5) [figure omitted; refer to PDF]

(6) If [figure omitted; refer to PDF] , then go to end

(7) Choose [figure omitted; refer to PDF]

(8) Let [figure omitted; refer to PDF]

(9) If there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , then go to step 6

(10) If there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , then go to step 6

(11) Print [figure omitted; refer to PDF]

(12) End

Algorithm 12.

(1) Input [figure omitted; refer to PDF]

(2) Let [figure omitted; refer to PDF]

(3) For all [figure omitted; refer to PDF] , let [figure omitted; refer to PDF]

(4) Input [figure omitted; refer to PDF] and [figure omitted; refer to PDF] satisfied in Algorithm 11

(5) Let [figure omitted; refer to PDF] ; for all [figure omitted; refer to PDF]

(6) [figure omitted; refer to PDF]

(7) If [figure omitted; refer to PDF] , then go to end

(8) Choose [figure omitted; refer to PDF]

(9) Let [figure omitted; refer to PDF]

(10) If there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , then go to step 7

(11) Let [figure omitted; refer to PDF]

(12) Let [figure omitted; refer to PDF] ; for all [figure omitted; refer to PDF]

(13) Choose [figure omitted; refer to PDF]

(14) Let [figure omitted; refer to PDF]

(15) If there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , then go to step 7

(16) If there is [figure omitted; refer to PDF] such that [figure omitted; refer to PDF] , then go to step 7

(17) Print two rings [figure omitted; refer to PDF] and [figure omitted; refer to PDF] with representations [figure omitted; refer to PDF] and [figure omitted; refer to PDF] , respectively, are isomorphic

(18) End

Example 13.

We compute the presentations of all finite rings of order [figure omitted; refer to PDF] , whose abelian group is isomorphic to [figure omitted; refer to PDF] . Let [figure omitted; refer to PDF] be a such ring. Then we write the following presentation for [figure omitted; refer to PDF] : [figure omitted; refer to PDF]

The outputs of these algorithms for the rings of order [figure omitted; refer to PDF] whose abelian group is [figure omitted; refer to PDF] are listed below. [figure omitted; refer to PDF]

Thus there exist [figure omitted; refer to PDF] rings of order [figure omitted; refer to PDF] with characteristic [figure omitted; refer to PDF] .

Acknowledgment

This research was supported by Islamic Azad University, Khorramabad Branch.

References

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[4] R. S. Wilson, "Representations of finite rings," Pacific Journal of Mathematics , vol. 53, no. 2, pp. 643-649, 1974.

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[6] R. Gilmer, J. Mott, "Associative rings of order ^{p 3} ," Proceedings of the Japan Academy , vol. 49, pp. 795-799, 1973.

[7] J. B. Derr, G. F. Orr, P. S. Peck, "Noncommutative rings of order ^{p 4} ," Journal of Pure and Applied Algebra , vol. 97, no. 2, pp. 109-116, 1994.

[8] B. Corbas, G. D. Williams, "Rings of order ^{p 5} . II. Local rings," Journal of Algebra , vol. 231, no. 2, pp. 691-704, 2000.

[9] M. Behboodi, R. Beyranvand, A. Hashemi, H. Khabazian, "Classification of finite rings: theory and algorithm," In press